How should we pool effect sizes in meta-analysis?
2024-07-30
To get our meta-analysis result, we need to take a weighted average of effect sizes…
But how should we weigh the evidence of each study?
Study weight \(w_{k}\) is inversely related to the variance (\(SE^2_{k}\))
\[ w_{k}=\frac{1}{(SE)^{2}_{k}} \]
The pooled estimate \(\widehat{\theta}\) is the weighted average:
\[ \widehat{\theta} = \frac{\Sigma_{k=1}^{K} \widehat{\theta}_{k} w_{k}}{\Sigma_{k=1}^{K} w_{k}} \]
A study’s effect size is an estimate of the true study effect size plus sampling error.
\[ \widehat{\theta}_{k}=\theta_{k} +\epsilon_{k} \]
The true effect size of study \(k\) is drawn from a distribution of effect sizes with mean \(\mu\), with error \(\zeta_{k}\).
\[ {\theta}_{k}=\mu+\zeta_{k} \]
\[ w^{*}_{k}=\frac{1}{SE^{2}_{k} + \tau^2} \]
Then compute the weighted average in the usual fashion.
Only choose the FEM if you have clear reason to (rare)
For most scenarios, use the REM.